An addendum to: Analytically Riesz operators and Weyl and Browder type theorems (original) (raw)

Filomat, 2014

A bounded linear operator A on a Banach space X is said to be ?polynomially Riesz?, if there exists a nonzero complex polynomial p such that the image p(A) is Riesz. In this paper we give some characterizations of these operators.

Schauder's theorem and Riesz theory for compact-like operators

Mathematische Zeitschrift, 1986

A well-known theorem of Schauder states that the transpose of a compact operator on a normed space is also a compact operator. This theorem has been generalized to locally convex spaces in several ways [9, w 42, pp. 200-204]. The purpose of the first part of this paper is to present a generalization of this result which is very close in spirit to Schauder's original theorem. Our proof even uses the same tools he used. This shows that the class of operators for which our result holds, a class which properly contains the compact operators, should be investigated further. We do that in the second part of this paper where the Riesz theory for our new class of operators is developed.

A note on Riesz spaces with property-b

Czechoslovak Mathematical Journal, 2006

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Some characterizations of Riesz spaces in the sense of strongly order bounded operators

Positivity, 2019

We investigate some properties of strongly order bounded operators. For example, we prove that if a Riesz space E is an ideal in E ∼∼ and F is a Dedekind complete Riesz space then for each ideal A of E, T is strongly order bounded on A if and only if T A is strongly order bounded. We show that the class of strongly order bounded operators satisfies the domination problem. On the other hand, we present two ways for decomposition of strongly order bounded operators, and we give some of their properties. Also, it is shown that E has order continuous norm or F has the b-property whenever each pre-regular operator form E into F is order bounded.

Carleman operators in Riesz spaces

Indagationes Mathematicae (Proceedings), 1983

A linear operator T from a normed space G into a Riesz space F is called a Carleman operator if the image of the unit ball in G is an order bounded subset of the universal completion of F. This abstract formulation is adequate to generalize the classical results concerning Carleman operators to the setting where no measure space is involved. This includes the characterizations of these operators, their relationship with abstract kernel operators and some compactness-type properties. In the special case where G and/or F are ideals of measurable functions, we regain the results of V.B. Korotkov and the recent results of A.R. Schep (Proc. Kon. Ned. Akad. Wetensch. A83 (Indag. Math. 42), 49-59 (1980)). We also show how some of the results of N.E. Gretsky and J.J. Uhl (Acta Sci. Math. (Szeged) 43, 207-218, 1981) fit into this general framework.